1. Axioms and rules of inference for propositional logic. Suppose T
... Proof. See page 81 of the coursepack for the proof of the first one. We leave the proof of the second one as an exercise for the reader. ...
... Proof. See page 81 of the coursepack for the proof of the first one. We leave the proof of the second one as an exercise for the reader. ...
Countable or Uncountable…That is the question!
... of the sequence a1, a2, a3, . . ., thus we may order the elements of B as b1, b2, b3, . . . where bk = ck and the function f (i) = bi is a bijection between N and B ...
... of the sequence a1, a2, a3, . . ., thus we may order the elements of B as b1, b2, b3, . . . where bk = ck and the function f (i) = bi is a bijection between N and B ...
Countable or Uncountable*That is the question!
... of the sequence a1, a2, a3, . . ., thus we may order the elements of B as b1, b2, b3, . . . where bk = ck and the function f (i) = bi is a bijection between N and B ...
... of the sequence a1, a2, a3, . . ., thus we may order the elements of B as b1, b2, b3, . . . where bk = ck and the function f (i) = bi is a bijection between N and B ...
PDF
... ` A → B, then ` B), all we need to show is that every axiom of PLi is a theorem of PLc . 1. A → (B → A). This is just an axiom schema for PLc . 2. A → (B → A ∧ B). A, B, A → ¬B, ¬B, ⊥ leads to A, B, A → ¬B `⊥. Applying the deduction theorem three times, we get ` A → (B → ((A → ¬B) →⊥)), or ` A → (B ...
... ` A → B, then ` B), all we need to show is that every axiom of PLi is a theorem of PLc . 1. A → (B → A). This is just an axiom schema for PLc . 2. A → (B → A ∧ B). A, B, A → ¬B, ¬B, ⊥ leads to A, B, A → ¬B `⊥. Applying the deduction theorem three times, we get ` A → (B → ((A → ¬B) →⊥)), or ` A → (B ...
Lecture Notes - Department of Mathematics
... Otherwise A is tame. Hence the set of all sets is wild. On the other hand, the set of real numbers or any other sets you encountered in calculus class are tame. Let Ω be the set of all tame sets. (Ω being the last letter of the Greek alphabet is often used to denote “large” sets.) Is Ω tame? If it i ...
... Otherwise A is tame. Hence the set of all sets is wild. On the other hand, the set of real numbers or any other sets you encountered in calculus class are tame. Let Ω be the set of all tame sets. (Ω being the last letter of the Greek alphabet is often used to denote “large” sets.) Is Ω tame? If it i ...
SORT LOGIC AND FOUNDATIONS OF MATHEMATICS 1
... aL (P )-tuples of elements of sorts n1 , . . . , nk , where (n1 , . . . , nk ) = s(P ). So we can read off from every n-ary predicate symbol what the sorts of the elements are in the n-tuples of the intended relation. In other words, we do not have symbols for abstract relations between elements of ...
... aL (P )-tuples of elements of sorts n1 , . . . , nk , where (n1 , . . . , nk ) = s(P ). So we can read off from every n-ary predicate symbol what the sorts of the elements are in the n-tuples of the intended relation. In other words, we do not have symbols for abstract relations between elements of ...
A Simple and Practical Valuation Tree Calculus for First
... that the reader pays a closer attention to the example given there. ...
... that the reader pays a closer attention to the example given there. ...
Running Time of Euclidean Algorithm
... the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no intersections. On the other hand, plugging n = 1 into n(n-1)/2 gi ...
... the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no intersections. On the other hand, plugging n = 1 into n(n-1)/2 gi ...
Strong Completeness and Limited Canonicity for PDL
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
Strong Logics of First and Second Order
... 2 we shall start by investigating two traditional strong logics (ω-logic and β-logic) that share many of these features of absoluteness, only now absoluteness is secured relative to ZFC. These logics will serve as our guide in setting up stronger logics that are absolute relative to stronger backgro ...
... 2 we shall start by investigating two traditional strong logics (ω-logic and β-logic) that share many of these features of absoluteness, only now absoluteness is secured relative to ZFC. These logics will serve as our guide in setting up stronger logics that are absolute relative to stronger backgro ...
Review - UT Computer Science
... respect to particular interpretations of interest. One example is Presburger arithmetic, in which the universe is the natural numbers and there is a single function, plus, whose properties are axiomatized. There are other theories that are incomplete because we have not yet added enough axioms. But ...
... respect to particular interpretations of interest. One example is Presburger arithmetic, in which the universe is the natural numbers and there is a single function, plus, whose properties are axiomatized. There are other theories that are incomplete because we have not yet added enough axioms. But ...
On Natural Deduction in Classical First-Order Logic: Curry
... kept as possible alternatives, since one is not able to decide which branch is going to be executed at the end. The informal idea expressed by the associated reductions is to assume ∀α P and try to produce some complete proof of C out of u by reducing inside u. Whenever u needs the truth of an insta ...
... kept as possible alternatives, since one is not able to decide which branch is going to be executed at the end. The informal idea expressed by the associated reductions is to assume ∀α P and try to produce some complete proof of C out of u by reducing inside u. Whenever u needs the truth of an insta ...
A SHORT AND READABLE PROOF OF CUT ELIMINATION FOR
... Montagna [7] showed that the “natural” first-order extension of GL, known as Quantified GL (QGL), is not as “nice”: Its Gentzenisation provably is not a provability logic (loc. cit.), but it even fails other nice properties such as cut elimination ([1]) and Craig interpolation; however it satisfies ...
... Montagna [7] showed that the “natural” first-order extension of GL, known as Quantified GL (QGL), is not as “nice”: Its Gentzenisation provably is not a provability logic (loc. cit.), but it even fails other nice properties such as cut elimination ([1]) and Craig interpolation; however it satisfies ...
Quantified Equilibrium Logic and the First Order Logic of Here
... present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternative axiom set for first-order here-and-there. The new system appears to be ...
... present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternative axiom set for first-order here-and-there. The new system appears to be ...
An Example of Induction: Fibonacci Numbers
... Theorem 1. The Fibonacci number F5n is a multiple of 5, for all positive integers n. Proof. Proof by induction on n. Since this is a proof by induction, we start with the base case of n = 1. That means, in this case, we need to compute F5×1 = F5 . But it is easy to compute (from the definition) that ...
... Theorem 1. The Fibonacci number F5n is a multiple of 5, for all positive integers n. Proof. Proof by induction on n. Since this is a proof by induction, we start with the base case of n = 1. That means, in this case, we need to compute F5×1 = F5 . But it is easy to compute (from the definition) that ...
x - Stanford University
... arguments, but each function has a fixed arity. Functions evaluate to objects, not propositions. There is no syntactic way to distinguish functions and predicates; you'll have to look at how they're used. ...
... arguments, but each function has a fixed arity. Functions evaluate to objects, not propositions. There is no syntactic way to distinguish functions and predicates; you'll have to look at how they're used. ...
On Action Logic
... a finite relation. Elements of Σ (resp. N ) are called terminal (resp. nonterminal) symbols of G. S is the initial symbol of G. P is the set of production rules of G. The terminal alphabet of G will also be denoted by ΣG . Let G be a CF-grammar, and x, y ∈ (Σ ∪ N )∗ . We say that y is directly deri ...
... a finite relation. Elements of Σ (resp. N ) are called terminal (resp. nonterminal) symbols of G. S is the initial symbol of G. P is the set of production rules of G. The terminal alphabet of G will also be denoted by ΣG . Let G be a CF-grammar, and x, y ∈ (Σ ∪ N )∗ . We say that y is directly deri ...
The Development of Mathematical Logic from Russell to Tarski
... the scholastics, set of the mathematicians, common noun of ordinary language”). The notion of class is not defined but assumed with its informal meaning. Extensionality for classes is also assumed: “a class is completely known when one knows which individuals belong to it.” However, the notion of or ...
... the scholastics, set of the mathematicians, common noun of ordinary language”). The notion of class is not defined but assumed with its informal meaning. Extensionality for classes is also assumed: “a class is completely known when one knows which individuals belong to it.” However, the notion of or ...
na.
... Cl (m). Notice that here we consider Cl (tJ) not a -(a.). One may see that ..A •s . is a pseudo-model for m, where s and e are defined below and e· is the retiexlve and transitive closure of B. The relation s is defined by DIs Da itt there is a g-consistent and complete theory T1 such that D1 T1 n C ...
... Cl (m). Notice that here we consider Cl (tJ) not a -(a.). One may see that ..A •s . is a pseudo-model for m, where s and e are defined below and e· is the retiexlve and transitive closure of B. The relation s is defined by DIs Da itt there is a g-consistent and complete theory T1 such that D1 T1 n C ...
YABLO WITHOUT GODEL
... or is true for. It is to be read: For all x and y, the formula ‘ϕ(x, y)’ is satisfied by x and y iff ϕ(x, y). An instance in English would be the following sentence: ‘is bigger than’ is satisfied by objects x and y iff x is bigger than y. The variables x and y are fixed; the first two variables in a ...
... or is true for. It is to be read: For all x and y, the formula ‘ϕ(x, y)’ is satisfied by x and y iff ϕ(x, y). An instance in English would be the following sentence: ‘is bigger than’ is satisfied by objects x and y iff x is bigger than y. The variables x and y are fixed; the first two variables in a ...
A Calculus for Belnap`s Logic in Which Each Proof Consists of Two
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
Definability in Boolean bunched logic
... A property P of BBI-models is said to be definable if there exists a formula A such that for all BBI-models M , A is valid in M ⇐⇒ M ∈ P. We’ll consider properties that feature in various models of separation logic. To show a property is definable, just exhibit the defining ...
... A property P of BBI-models is said to be definable if there exists a formula A such that for all BBI-models M , A is valid in M ⇐⇒ M ∈ P. We’ll consider properties that feature in various models of separation logic. To show a property is definable, just exhibit the defining ...
THE EQUALITY OF ALL INFINITIES
... By the “real line”, we mean a line on which we marked some point as zero. Then at equal intervals to the right of 0 , we mark the counting numbers ...
... By the “real line”, we mean a line on which we marked some point as zero. Then at equal intervals to the right of 0 , we mark the counting numbers ...
The Emergence of First
... in which he could have defined the general notion of cardinal number as he did, deriving it from logic, without the use of second-order logic. In the Fundamental Laws (1893), Frege also introduced a hierarchy of levels of quantification. After discussing first-order and second-order propositional fu ...
... in which he could have defined the general notion of cardinal number as he did, deriving it from logic, without the use of second-order logic. In the Fundamental Laws (1893), Frege also introduced a hierarchy of levels of quantification. After discussing first-order and second-order propositional fu ...
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set ""number"". The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the ""underlying logic"". The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.